Decision making relative to process changes using simulation, including how to create and report the variability of NPV.

Confidence interval: probability that the interval around the sample mean will contain the actual population mean.
When population SD is not known: x bar +/- tα/2* s/sqrt of n
When population SD is known: xbar +/- Zα/2*σ/sqrt of n
If we could say that the mean of the sample equaled the mean of the population (we can onlybe 95% confident that the population mean even falls in the 95% CI of the sample mean) and the distribution is normal, then the probability of being at a given value or less (or more) when we are dealing with a normal distribution is:
If you have the population standard deviation, then: (value-sample mean)/σ = Z, use z to determine p of being the value or beyond
If you just have a sample and you assume it reflects the population (big assumption, why use means) then: value-mean /s=t, use t to determine p of being the value or beyond. A sample of 1000 using modeling to generate samples would probably be safe to consider that the population is represented, that is why the t score and z score converge after about 200 measures.
If the distribution of the sample is not normal, all distributions have a cumulative density function that enables an analyst to make the same statement about a given area within the distribution. Excel has functions for the normal, binomial, chi square, exponential, F distribution, gamma, log, Weibull, and the beta distributions. The formulas and template you need to use the triangular distribution are presented in the file ‘Triangular Distribution Formulas’, just put in the min, mode, and max in the appropriate cells and then in a value in the ‘x’ cell that is within the min to max range.
To determine a CI you need:

  1. An adequate size sample that is normally distributed
  2. The more variation within the model and displayed in the output, the larger the NPV sample. If the SD of the outputs used to generate the payment and the NPV is less than 10% of the sample mean, let us say that the 30 samples generated by Arena when we ask for 30 iterations are adequate. Otherwise, use Monte Carlo simulation to generate 1000 possible values of each variable used to generate the payment by pulling from the distribution that Input Analyzer indicates.
  3. If the distribution of the NPV’s is normally distributed, than just use that set as the data used to generate the CI. Use Input Analyzer to determine if the distribution is normal. If it is not, then bootstrap. Generate 30 samples of 10 data points each to determine the mean of means and their standard deviation so you can determine the CI (central limit theorem).
  4. The mean
  5. The SD of the sample

So what can a person say about what the NPV will be if the system is changed? We can say that the sample we used to generate the CI indicates that the real mean of the population has a 95% probability of being within that CI. Of course the population in this case is one NPV that we get when we change the system, we do not have a time machine to go back and repeat the time interval used to generate the NPV an infinite number of times to generate a population of possibilities and a population mean. Thus, we might want to inform the manager of the p of being less than some number near the min (min when you have a closed distribution, min and zero if you have a continuous distribution) of the sample used to generate the CI. Then put in the caveat that the CI, the figure and p of being less than that figure only makes sense for the model and output generated, if the base variabilities are wrong or change dramatically (perform sensitivity analysis to determine what dramatically is)then the information will not be an accurate representation of the actual outcome.